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Electrostatic solution of massless quenches in Luttinger liquids

by Paola Ruggiero, Pasquale Calabrese, Thierry Giamarchi, Laura Foini

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Laura Foini · Thierry Giamarchi · Paola Ruggiero
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-09-21
Date submitted: 2022-05-22 18:54
Submitted by: Ruggiero, Paola
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


The study of non-equilibrium dynamics of many-body systems after a quantum quench received a considerable boost and a deep theoretical understanding from the path integral formulation in imaginary time. However, the celebrated problem of a quench in the Luttinger parameter of a one dimensional quantum critical system (massless quench) has so far only been solved in the real-time Heisenberg picture. In order to bridge this theoretical gap and to understand on the same ground massive and massless quenches, we study the problem of a gaussian field characterized by a coupling parameter K within a strip and a different one K0 in the remaining two semi-infinite planes. We give a fully analytical solution using the electrostatic analogy with the problem of a dielectric material within a strip surrounded by an infinite medium of different dielectric constant, and exploiting the method of charge images. After analytic continuation, this solution allows us to obtain all the correlation functions after the quench within a path integral approach in imaginary time, thus recovering and generalizing the results in real time. Furthermore, this imaginary-time approach establishes a remarkable connection between the quench and the famous problem of the conductivity of a Tomonaga-Luttinger liquid coupled to two semi-infinite leads: the two are in fact related by a rotation of the spacetime coordinates.

Author comments upon resubmission

Dear Editors,

We are submitting a revised version of the manuscript
“Electrostatic solution of massless quenches in Luttinger liquids”.

We would like to thank the editors for their work and the referees for
their useful comments and suggestions.

We believe that we thoroughly addressed all the comments of the referees
in the new version of the manuscript.


Paola Ruggiero
Pasquale Calabrese
Thierry Giamarchi
Laura Foini

List of changes

- Few typos corrected
- New section added (Section 7)
- Last sentence in Appendix added

Published as SciPost Phys. 13, 111 (2022)

Reports on this Submission

Anonymous Report 3 on 2022-8-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2203.06740v3, delivered 2022-08-30, doi: 10.21468/SciPost.Report.5605


The paper fills a gap compared to previous works, by proposing to use the imaginary-time path integral for massless quenches in Luttinger Liquids
It makes clear the analogy with the spatial inhomogeneous Luttinger liquid, obtained by a permutation of space and time coordinates.


The paper recovers an already know result (Eq.29) obtained in Refs. 29,50-52. It is not so clear to guess whether the different method exposed here could give access to other unsolved problems.


The paper is based on the path integral formulation for massless quenches in Luttinger liquids. The main interest of the paper resides into an alternative method to those used in Refs. 29,50-52 to recover the same result. The starting formal expression that allows to switch from real to imaginary time , given by Eq. 4, corresponds to Eq.2 in Ref.5 for instance, where it was developed for massive quenches. Its limitations was addressed in Ref. 6, page 4: "Unfortunately, in confined geometries, only a few field theories with specific boundary conditions can be solved analytically in such a way to have results that can be continued from real to complex values". If I understand well, the present situation corresponds to such an example, thus could be valuable in this respect (maybe the authors could confirm or not this point?). Nonetheless, I have hesitated as it is not clear whether the present method could solve other problems such as those mentioned in the conclusion. The pedagogical value of the paper incites me to recommend its publication in SciPost. The connexion to spatially inhomogeneous Luttinger liquid is also interesting.

Requested changes

I have some remarks about the way similarity to the spatially inhomogeneous Luttinger liquid is presented. Either in the introduction or section 6, authors insist only on the motivation and result for the dc conductance, while the finite frequency Green's function, related to the non-local conductivity, has been obtained (first) in Ref. 38 (and not in Ref.39). The present equation 39 corresponds precisely to Eq.9 in Ref.38. It would be interesting to provide the interpretation thus given of this equation, due to multiple reflexions inside the wire which acts a a Fabry-Perot resonator. By developing the scattering approach to plasmons, the ac conductance was related exactly to the transmission coefficient constructed through evolution of plasmonic modes. As the sum of all transmitted spikes sums up to one (and the sum of reflected ones sum up to zero), the dc conductance is equal to e^2/h$. By the way, it is stated in the introduction (page 2) that "as soon as we consider an inhomogeneous Luttinger parameter K(y,z)... the propagator has to be determined numerically". This is not clearly the case in Refs.38,53,54 which have provided an analytical solution for the propagator.

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

Anonymous Report 2 on 2022-6-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2203.06740v3, delivered 2022-06-05, doi: 10.21468/SciPost.Report.5184


I thank the authors for their reply to my questions about the manuscript. As stated in my previous report, the manuscript is scientifically correct and well-written. It elaborates on the study of the massless quench in Luttinger liquids, which has a broad interest for the community working on the non-equilibuim dynamics after a quench. On a more technical side, it gives an exact analytic solution for the Green's function in an inhomogeneous medium. Despite the simple form of the inhomogeneity of the Luttinger parameter, this result is quite remarkable. For these reasons, I recommend this manuscript for the publication in SciPost Physics.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2022-5-27 (Invited Report)


I thank the authors for their detailed response. Unfortunately they appear to have misunderstood my main point, which was not that the whole approach using imaginary time is narrow and pedestrian, but that this particular contribution, although correct, is so, and is not at the level suitable for publication in SciPost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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